As I said, the rule is kind of weird, and I have no obvious geometric interpretation to offer for it. It does, however, satisfy the expected properties that:

When the unit is exactly in the middle of any cell X, then (regardless of which of A, B, C or D we choose X to be) it contributes 1λ = 1 unit of density to cell X and 0 units of density to any other cell.

When the unit is exactly at the corner of four cells, it contributes (1/2)λ units of density to each of the four cells (and nothing to any other cell, by definition).

However, note that, using this rule, the total amount of density contributed by a unit to all cells is not constant, not even if λ = 1. In particular, when the unit is exactly at the midpoint of the edge between two cells, it contributes (1/2)λ units of density to those two cells, and nothing to any other cell. Thus, calling the resulting value a "density" seems a bit misleading.

Edit: Another way of writing the formulas for ρA, ρB, ρC and ρD, which may make the symmetry of the definitions more apparent, is to define dX = max( |x − xX|, |y − yX| ) as the chessboard distance of the unit at (x, y) from the center of the cell X at (xX, yX), measured in cell widths/heights.
Then, for any cell X,